Non-locality of Symmetric States

Damian Markham

Joint work with:Zizhu Wang

CNRS, LTCIENST (Telecom ParisTech), Paris

quant-ph/1112.3695

Motivation

• Non-locality as an interpretation/witness for entanglement ‘types’?

(mutlipartite entanglment is a real mess)

• Different ‘types’ of non-locality?(is non-locality also such a mess?)

Look at symmetric states – same tool ‘Majorana Representation’ used to study both….

Outline1. Background (entanglement classes, non-locality,

Majorana representation for symmetric states)

2. Hardy’s Paradox for symmetric states of n-qubits

3. Different Hardy tests for different entanglement classes

Entanglement

Definition:

State is entangled iff NOT separapable

SEP

entangled

Entanglement

Types of entanglement Dur, Vidal, Cirac, PRA 62, 062314 (2000) In multipartite case, some states are incomparable, even under stochastic Local Operations and Classical Communications (SLOCC)

Infinitely many different classes!

• Different resources for quantum information processing

• Different entanglement measures may apply for different types

1 2SLOCC

Permutation Symmetric States

Symmetric under permutation of parties

• Occur as ground states e.g. of some Bose Hubbard models

• Useful in a variety of Quantum Information Processing tasks

• Experimentally accessible in variety of media

ijP

nH

Permutation Symmetric States Majorana representationE. Majorana, Nuovo Cimento 9, 43 – 50 (1932)

cos( / 2) 0 sin( / 2) 1iii i ie

12

3

4

Permutation Symmetric States Majorana representationE. Majorana, Nuovo Cimento 9, 43 – 50 (1932)

2 /32 1/ 2 0 1ie

12

3 1 1/ 2 0 1

4 /32 1/ 2 0 1ie

Permutation Symmetric States Majorana representationE. Majorana, Nuovo Cimento 9, 43 – 50 (1932)

Dicke states

n k

k

Permutation Symmetric States Majorana representationE. Majorana, Nuovo Cimento 9, 43 – 50 (1932)

35

1

22

23

4

• Distribution of points alone determines entanglement features

Local unitary rotation of sphere

• Orthogonality relations

0n

i

Permutation Symmetric States

• Different degeneracy classes* ( )

• Different symmetries^

*T. Bastin, S. Krins, P. Mathonet, M. Godefroid, L. Lamata and E. Solano , PRL 103, 070503 (2009) ^D.M. PRA (2010)

id

4GHZ 4(4,1)S W

Different entanglement ‘types’ (w.r.t. SLOCC)

Permutation Symmetric States

• Different degeneracy classes ( )

• Different symmetries

• NOTE: Almost identical states can be in different classes

id

4GHZ

Different entanglement ‘types’ (w.r.t. SLOCC)

Comparison to Spinor BEC

E.g. S=2R. Barnett, A. Turner and E. Demler, PRL 97, 180412 (2007)

Phase diagram for spin 2 BEC in single optical trap(Fig taken from PRL 97, 180412)

Comparison to Spinor BEC

E.g. S=2R. Barnett, A. Turner and E. Demler, PRL 97, 180412 (2007)

Phase diagram for spin 2 BEC in single optical trap(Fig taken from PRL 97, 180412)

Non-Locality

0,1a 0,1b

0,1A 0,1B

Measurement basis

Measurement outcome

V

H 'H

'V

0a 1a

Non-Locality

• Local Hidden Variable model

0,1a 0,1b

0,1A 0,1B

Measurement basis

Measurement outcome

Non-Locality

• Local Hidden Variable model

1 0,1m

1 0,1M

Measurement basis

Measurement outcome

2 0,1m

2 0,1M

0,1nm

0,1nM

n Party case (Hardy’s Paradox)

• For all symmetric states of n qubits (except Dicke states)

- set of probabilities which contradict LHV- set of measurement which achieve these probabilities

measurement basis

measurement result

n Party case (Hardy’s Paradox)

• For all symmetric states of n qubits (except Dicke states)

- set of probabilities which contradict LHV- set of measurement which achieve these probabilities

measurement basis

measurement result

for some ,

n Party case (Hardy’s Paradox)

• For all symmetric states of n qubits (except Dicke states)

- set of probabilities which contradict LHV- set of measurement which achieve these probabilities

measurement basis

measurement result

for some ,

for same ,

n Party case (Hardy’s Paradox)

• For all symmetric states of n qubits (except Dicke states)

- set of probabilities which contradict LHV- set of measurement which achieve these probabilities

measurement basis

measurement result

for some ,

for same ,

CONTRADICTION!

n Party case (Hardy’s Paradox)

• For all symmetric states of n qubits (except Dicke states)

- set of probabilities which contradict LHV- set of measurement which achieve these probabilities

measurement basis

measurement result

n Party case (Hardy’s Paradox)

• Use Majorana representation

n Party case (Hardy’s Paradox)

• Use Majorana representation

11

n Party case (Hardy’s Paradox)

11

?

• Majorana representation always allows satisfaction of lower conditions, what about the top condition?

- Must find, such that

n Party case (Hardy’s Paradox)

11

• Majorana representation always allows satisfaction of lower conditions, what about the top condition?

- Must find, such that

- It works for all cases except product or Dicke states Dicke states too symmetric!

(almost)

All permutation symmetric states are non-local!

Testing Entanglement class

• Use the Majorana representation to add constraints implying degeneracy

0n

i

0i

1in d

0i

1in d

Testing Entanglement class

Testing Entanglement class

• Use the Majorana representation to add constraints implying degeneracy

0n

i

0i

1in d

0i

1in d

• Use the Majorana representation to add constraints implying degeneracy

0n

i

0i

1in d

0i

1in d

Only satisfied by With degeneracy

id di

Testing Entanglement class

• Use the Majorana representation to add constraints implying degeneracy

• Correlation persists to fewer sets, only for degenerate states.

Only satisfied by With degeneracy

id di

1

i k

ii

d n

1

1 ii k

di

iPermK

Testing Entanglement class

e.g. Hardy Test for W class…

TW

Get REAL!!!

• We really need to talk about inequalities

• ALL states with one MP of degeneracy or greater violate for

• CanNOT be any longer true that ‘only’ states of the correct type violate

- states arbitrarily close which are in different class

Get REAL!!!

Persistency of degenerate states

4(4,1)S W

Conclusions

• Hardy tests for almost all symmetric states

• Different entanglement types Different Hardy tests

- Non-locality to ‘witness’ / interpret entanglement types- Different flavour of non-local arguments to Stabiliser states/

GHZ states e.t.c.

• More?- ‘Types’ of non-locality? (from operational perspective)- Witness phase transitions by different locality tests?- Meaning in terms of non-local games?

Thank you!

http://iq.enst.fr/

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